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On the Brauer groups of fibrations II

Yanshuai Qin

Abstract

Let $K$ be a number field, and let $\mathcal{X}$ be a proper regular flat scheme over $\mathcal{O}_{K}$ with a generic fiber $X$ geometrically connected over $K$. We prove that there is an exact sequence up to finite groups $0\rightarrow Sha(Pic_{X/K}^0)\rightarrow Br(\mathcal{X})\rightarrow Br(X_{\bar{K}})^{G_K}\rightarrow 0$, which generalizes a theorem of Artin and Grothendieck for arithmetic surfaces to arbitrary dimensions. Consequently, we reduce Artin's question regarding the finiteness of $Br(\mathcal{X})$ for proper regular flat schemes $\mathcal{X}$ over $\mathbb{Z}$ to $3$-dimensional arithmetic schemes.

On the Brauer groups of fibrations II

Abstract

Let be a number field, and let be a proper regular flat scheme over with a generic fiber geometrically connected over . We prove that there is an exact sequence up to finite groups , which generalizes a theorem of Artin and Grothendieck for arithmetic surfaces to arbitrary dimensions. Consequently, we reduce Artin's question regarding the finiteness of for proper regular flat schemes over to -dimensional arithmetic schemes.

Paper Structure

This paper contains 18 sections, 42 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Sketch of Proofs
  4. Sketch of the proof of Proposition \ref{['cor1']}
  5. The idea of the proof of Theorem \ref{['mainthm']}
  6. Notation and Terminology
  7. Preliminary
  8. Local results
  9. Local invariant cycle theorems
  10. Proof of Theorem \ref{['main']}
  11. Proof of Proposition \ref{['cor1']}
  12. Proof of Corollary \ref{['big']}
  13. Removing the semi-stable assumption
  14. Proof of Theorem \ref{['0cycle']}
  15. Proof of Theorem \ref{['mainthm']}
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $\pi:\mathcal{X}\longrightarrow C$ be a proper flat morphism, where $C$ is $\mathop{\mathrm{Spec}}\nolimits {\mathcal{O}}_K$ for some number field $K$. Assume that $\mathcal{X}$ is regular and the generic fiber $X$ of $\pi$ is projective and geometrically connected over $K$. Then there is an exa

Theorems & Definitions (92)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 1.7
  • Remark 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 82 more